let x0, y0, r be Real; :: thesis: for z being Element of REAL 2

for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds

( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds

( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) ) )

assume that

A1: z = <*x0,y0*> and

A2: f is_partial_differentiable_in z,2 ; :: thesis: ( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) ; :: thesis: r = partdiff (f,z,2)

r = diff ((SVF1 (2,f,z)),y0) by A1, A2, A3, Lm2;

hence r = partdiff (f,z,2) by A1, Th2; :: thesis: verum

for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds

( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds

( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) ) )

assume that

A1: z = <*x0,y0*> and

A2: f is_partial_differentiable_in z,2 ; :: thesis: ( r = partdiff (f,z,2) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )

hereby :: thesis: ( ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) implies r = partdiff (f,z,2) )

given x1, y1 being Real such that A3:
( z = <*x1,y1*> & ex N being Neighbourhood of y1 st ( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) implies r = partdiff (f,z,2) )

assume
r = partdiff (f,z,2)
; :: thesis: ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )

then r = diff ((SVF1 (2,f,z)),y0) by A1, Th2;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) by A1, A2, Lm2; :: thesis: verum

end;( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )

then r = diff ((SVF1 (2,f,z)),y0) by A1, Th2;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) by A1, A2, Lm2; :: thesis: verum

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) ; :: thesis: r = partdiff (f,z,2)

r = diff ((SVF1 (2,f,z)),y0) by A1, A2, A3, Lm2;

hence r = partdiff (f,z,2) by A1, Th2; :: thesis: verum